Prove that similar matrices have the same rank

  • Thread starter zeion
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  • #1
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Homework Statement



Prove that similar matrices have the same rank.


Homework Equations





The Attempt at a Solution



Similar matrices are related via: B = P-1AP, where B, A and P are nxn matrices..
since P is invertible, it rank(P) = n, and so since the main diagonal of P all > 0, multiplying by P will not change the rank of A, so rank B = rank A.

Is that seem right?
 

Answers and Replies

  • #2
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If you can show that rank(P-1AP) is less than or equal to rank(A), then you are done since matrix similarity is symmetric (if A is similar to B, then B is similar to A).
 

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